We propose a HfZrO _x (HZO)-based ferroelectric synapse device with multi-levels states of remnant polarization that is equivalent to multi-levels conductance states. By optimizing the pulse condition, we obtained 32 levels of remnant polarization states for both potentiation and depression. Furthermore, a ferroelectric field-effect transistor is simulated using the obtained multiple remnant polarization states. The simulation results show that linear and symmetric conductance states can be obtained by applying optimum potentiation and depression pulse conditions. A neural network was simulated using the proposed devices for pattern recognition. Using synapse parameters of the HZO-based ferroelectric device and a neural network simulator, we have confirmed that the pattern recognition accuracy of the MNIST data set is 84%. It shows that the HZO-based synapse device has potential for future high-density neuromorphic systems.